Find the products AB and BA to determine whether B is the multiplicative inverse of A.?

I don't know why we have to find the product of AB and then BA. My work is use Gauss-Jordan method to figure out inverse of A and then compare to B. and |dw:1360901929122:dw| is inverse of A, it is not a multiply matrix of B .

This uses the definition of the multiplicative inverse.
ie
if and only if,
$$AB=BA=I$$
A is the multiplicative inverse of B
So find the values of AB and BA and see whether they are equal to identity matrix

OK, the matrix with 1's on the diagonal is the identity matrix (called I (eye))
a * I will give you a
also, if you know
a * b = I
then you know b is the *inverse* of a
you also know b*a= I (but I think they want you to multiply it out and see that it is)
and we could just as well say a is the *inverse* of b
\[ A^{-1} A = I \]
(People use capital letters for matrices (bold face if you can). the use lower case, bold letters for vectors)

you mean "wouldn't it be
\[ B = A^{-1} \]
the -1 is not an exponent, but means *inverse*
You can say that. But the inverse of the inverse
\[ (A^{-1})^{-1} = A\]
if we take the inverse of both sides
\[ B^{-1} = (A^{-1})^{-1} = A \]
or
\[ A = B^{-1} \]
which says A is the inverse of B
(or vice versa)